A harmonic is any member of the harmonic series, a divergent infinite
series. Its name derives from the concept of overtones, or harmonics
in musical instruments: the wavelengths of the overtones of a
vibrating string or a column of air (as with a tuba) are derived from
the string's (or air column's) fundamental wavelength. Every term of
the series (i.e., the higher harmonics) after the first is the
"harmonic mean" of the neighboring terms. The phrase "harmonic mean"
likewise derives from music.
The term is employed in various disciplines, including music, physics,
acoustics, electronic power transmission, radio technology, and other
fields. It is typically applied to repeating signals, such as
sinusoidal waves. A harmonic of such a wave is a wave with a frequency
that is a positive integer multiple of the frequency of the original
wave, known as the fundamental frequency. The original wave is also
called the 1st harmonic, the following harmonics are known as higher
harmonics. As all harmonics are periodic at the fundamental frequency,
the sum of harmonics is also periodic at that frequency. For example,
if the fundamental frequency is 50 Hz, a common AC power supply
frequency, the frequencies of the first three higher harmonics are
100 Hz (2nd harmonic), 150 Hz (3rd harmonic), 200 Hz
(4th harmonic) and any addition of waves with these frequencies is
periodic at 50 Hz.

An nth characteristic mode, for n > 1, will have nodes that are not
vibrating. For example, the 3rd characteristic mode will have nodes at

1
3

displaystyle tfrac 1 3

L and

2
3

displaystyle tfrac 2 3

L, where L is the length of the string. In fact, each nth
characteristic mode, for n a multiple of 3, will not have nodes at
these points. These other characteristic modes will be vibrating at
the positions

1
3

displaystyle tfrac 1 3

L and

2
3

displaystyle tfrac 2 3

L. If the player gently touches one of these positions, then these
other characteristic modes will be suppressed. The tonal harmonics
from these other characteristic modes will then also be suppressed.
Consequently, the tonal harmonics from the nth characteristic modes,
where n is a multiple of 3, will be made relatively more prominent.[1]

In music, harmonics are used on string instruments and wind
instruments as a way of producing sound on the instrument,
particularly to play higher notes and, with strings, obtain notes that
have a unique sound quality or "tone colour". On strings, harmonics
that are bowed have a "glassy", pure tone. On stringed instruments,
harmonics are played by touching (but not fully pressing down the
string) at an exact point on the string while sounding the string
(plucking, bowing, etc.); this allows the harmonic to sound, a pitch
which is always higher than the fundamental frequency of the string.

Terminology[edit]
Harmonics may also be called "overtones", "partials" or "upper
partials". The difference between "harmonic" and "overtone" is that
the term "harmonic" includes all of the notes in a series, including
the fundamental frequency (e.g., the open string of a guitar). The
term "overtone" only includes the pitches above the fundamental. In
some music contexts, the terms "harmonic", "overtone" and "partial"
are used fairly interchangeably.
Characteristics[edit]

A whizzing, whistling tonal character, distinguishes all the harmonics
both natural and artificial from the firmly stopped intervals;
therefore their application in connection with the latter must always
be carefully considered.[2]

Most acoustic instruments emit complex tones containing many
individual partials (component simple tones or sinusoidal waves), but
the untrained human ear typically does not perceive those partials as
separate phenomena. Rather, a musical note is perceived as one sound,
the quality or timbre of that sound being a result of the relative
strengths of the individual partials. Many acoustic oscillators, such
as the human voice or a bowed violin string, produce complex tones
that are more or less periodic, and thus are composed of partials that
are near matches to integer multiples of the fundamental frequency and
therefore resemble the ideal harmonics and are called "harmonic
partials" or simply "harmonics" for convenience (although it's not
strictly accurate to call a partial a harmonic, the first being real
and the second being ideal).
Oscillators that produce harmonic partials behave somewhat like
one-dimensional resonators, and are often long and thin, such as a
guitar string or a column of air open at both ends (as with the modern
orchestral transverse flute).
Wind instrumentsWind instruments whose air column is
open at only one end, such as trumpets and clarinets, also produce
partials resembling harmonics. However they only produce partials
matching the odd harmonics, at least in theory. The reality of
acoustic instruments is such that none of them behaves as perfectly as
the somewhat simplified theoretical models would predict.
Partials whose frequencies are not integer multiples of the
fundamental are referred to as inharmonic partials. Some acoustic
instruments emit a mix of harmonic and inharmonic partials but still
produce an effect on the ear of having a definite fundamental pitch,
such as pianos, strings plucked pizzicato, vibraphones, marimbas, and
certain pure-sounding bells or chimes. Antique singing bowls are known
for producing multiple harmonic partials or multiphonics. [3] [4]
Other oscillators, such as cymbals, drum heads, and other percussion
instruments, naturally produce an abundance of inharmonic partials and
do not imply any particular pitch, and therefore cannot be used
melodically or harmonically in the same way other instruments can.
Partials, overtones, and harmonics[edit]
An overtone is any partial higher than the lowest partial in a
compound tone. The relative strengths and frequency relationships of
the component partials determine the timbre of an instrument. The
similarity between the terms overtone and partial sometimes leads to
their being loosely used interchangeably in a musical context, but
they are counted differently, leading to some possible confusion. In
the special case of instrumental timbres whose component partials
closely match a harmonic series (such as with most strings and winds)
rather than being inharmonic partials (such as with most pitched
percussion instruments), it is also convenient to call the component
partials "harmonics" but not strictly correct (because harmonics are
numbered the same even when missing, while partials and overtones are
only counted when present). This chart demonstrates how the three
types of names (partial, overtone, and harmonic) are counted (assuming
that the harmonics are present):

Frequency
Order
Name 1
Name 2
Name 3
WaveWave Representation
Molecular Representation

1 × f = 0440 Hz
n = 1
1st partial
fundamental tone
1st harmonic

2 × f = 0880 Hz
n = 2
2nd partial
1st overtone
2nd harmonic

3 × f = 1320 Hz
n = 3
3rd partial
2nd overtone
3rd harmonic

4 × f = 1760 Hz
n = 4
4th partial
3rd overtone
4th harmonic

In many musical instruments, it is possible to play the upper
harmonics without the fundamental note being present. In a simple case
(e.g., recorder) this has the effect of making the note go up in pitch
by an octave, but in more complex cases many other pitch variations
are obtained. In some cases it also changes the timbre of the note.
This is part of the normal method of obtaining higher notes in wind
instruments, where it is called overblowing. The extended technique of
playing multiphonics also produces harmonics. On string instruments it
is possible to produce very pure sounding notes, called harmonics or
flageolets by string players, which have an eerie quality, as well as
being high in pitch. Harmonics may be used to check at a unison the
tuning of strings that are not tuned to the unison. For example,
lightly fingering the node found halfway down the highest string of a
cello produces the same pitch as lightly fingering the node ​1⁄3
of the way down the second highest string. For the human voice see
OvertoneOvertone singing, which uses harmonics.
While it is true that electronically produced periodic tones (e.g.
square waves or other non-sinusoidal waves) have "harmonics" that are
whole number multiples of the fundamental frequency, practical
instruments do not all have this characteristic. For example, higher
"harmonics"' of piano notes are not true harmonics but are "overtones"
and can be very sharp, i.e. a higher frequency than given by a pure
harmonic series. This is especially true of instruments other than
stringed or brass/woodwind ones, e.g., xylophone, drums, bells etc.,
where not all the overtones have a simple whole number ratio with the
fundamental frequency. The fundamental frequency is the reciprocal of
the period of the periodic phenomenon.[5]
On stringed instruments[edit]

Playing a harmonic on a string

Harmonics may be singly produced [on stringed instruments] (1) by
varying the point of contact with the bow, or (2) by slightly pressing
the string at the nodes, or divisions of its aliquot parts (

1
2

displaystyle tfrac 1 2

,

1
3

displaystyle tfrac 1 3

,

1
4

displaystyle tfrac 1 4

, etc.). (1) In the first case, advancing the bow from the usual place
where the fundamental note is produced, towards the bridge, the whole
scale of harmonics may be produced in succession, on an old and highly
resonant instrument. The employment of this means produces the effect
called 'sul ponticello.' (2) The production of harmonics by the slight
pressure of the finger on the open string is more useful. When
produced by pressing slightly on the various nodes of the open strings
they are called 'Natural harmonics.' ... Violinists are well aware
that the longer the string in proportion to its thickness, the greater
the number of upper harmonics it can be made to yield.
— Grove's Dictionary of
MusicMusic and Musicians (1879)[6]

The following table displays the stop points on a stringed instrument,
such as the guitar (guitar harmonics), at which gentle touching of a
string will force it into a harmonic mode when vibrated. String
harmonics (flageolet tones) are described as having a "flutelike,
silvery quality" that can be highly effective as a special color or
tone color (timbre) when used and heard in orchestration.[7] It is
unusual to encounter natural harmonics higher than the fifth partial
on any stringed instrument except the double bass, on account of its
much longer strings.[8] Harmonics are widely used in plucked string
instruments, such as acoustic guitar, electric guitar and electric
bass. On an electric guitar played loudly through a guitar amplifier
with distortion, harmonics are more sustained and can be used in
guitar solos. In the heavy metal music lead guitar style known as
shred guitar, harmonics, both natural and artificial, are widely used.

Table of harmonics of a stringed instrument with colored dots
indicating which positions can be lightly fingered to generate just
intervals up to the 7th harmonic

Artificial harmonics[edit]
Although harmonics are most often used on open strings (natural
harmonics), occasionally a score will call for an artificial harmonic,
produced by playing an overtone on an already stopped string. As a
performance technique, it is accomplished by using two fingers on the
fingerboard, the first to shorten the string to the desired
fundamental, with the second touching the node corresponding to the
appropriate harmonic. On fretted instruments, such as an electric
guitar, the performer can look at the frets to determine where to stop
the string and where to touch the node. On unfretted instruments, such
as the violin and related instruments, playing artificial harmonics is
an advanced technique, as it requires the performer to find two
precise locations on the same string.
Other information[edit]
Harmonics may be either used or considered as the basis of just
intonation systems. Composer
Arnold Dreyblatt is able to bring out
different harmonics on the single string of his modified double bass
by slightly altering his unique bowing technique halfway between
hitting and bowing the strings. Composer
Lawrence BallLawrence Ball uses harmonics
to generate music electronically.
See also[edit]

Demonstration of 16 harmonics using electronic sine tones, starting
with 110 Hz fundamental, 0.5s each. Note that each harmonic is
presented at the same signal level as the fundamental; the sample
tones sound louder as they increase in frequency